PastExamLabPastExamLab

成功大學 95 年度 微積分

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第 1 題15 分
Compute the following limit:
(a)10 分
lim⁡n→∞1n∑i=1n(3in)3+(3in)4\lim\limits_{n \to \infty} \frac{1}{n} \sum\limits_{i=1}^{n} \sqrt{\left(\frac{3i}{n}\right)^3 + \left(\frac{3i}{n}\right)^4}.
(b)5 分
lim⁡x→1(1ln⁡x−1x−1)\lim\limits_{x \to 1} \left(\frac{1}{\ln x} - \frac{1}{x-1}\right).
第 2 題10 分
Let f(x)=∫1exln⁡(x+t)x4+x2t+5dtf(x) = \int_{1}^{e^x} \frac{\ln(x+t)}{\sqrt{x^4 + x^2 t + 5}} dt defined on (−1,∞)(-1, \infty).
(a)5 分
Show that ff is a strictly increasing function.
(b)5 分
Find (f−1)′(0)(f^{-1})'(0).
第 3 題10 分
Compute the improper integral ∫01x−2/3ln⁡x dx\int_{0}^{1} x^{-2/3} \ln x \, dx.
第 4 題10 分
Find the radius of convergence of the infinite series ∑n=0∞(n2n+1)nxn\sum\limits_{n=0}^{\infty} \left(\frac{n}{2n+1}\right)^n x^n.
第 5 題10 分
Let ff be a real valued function defined on R\mathbb{R} with f′′(x)>0f''(x) > 0 for all xx. Show that f(x)≥f′(0)x+f(0)f(x) \geq f'(0)x + f(0) for all xx.
第 6 題10 分
Compute the iterated integral ∫01∫01−x2sin⁡(x2+y2) dy dx\int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \sin(x^2 + y^2) \, dy \, dx.
第 7 題10 分
Find the local extreme values of the function f(x,y)=xy+1x+8yf(x, y) = xy + \frac{1}{x} + \frac{8}{y} for xy≠0xy \neq 0.
第 8 題10 分
Use the method of Lagrange Multiplier to find the maximum value of the function f(x,y,z)=x+3y−2zf(x, y, z) = x + 3y - 2z defined on R3\mathbb{R}^3 subject to the constraint x2+y2+z2=14x^2 + y^2 + z^2 = 14.
第 9 題15 分
Find the area of the region Ω\Omega bounded by the curves x2−4xy+4y2−2x−y−1=0x^2 - 4xy + 4y^2 - 2x - y - 1 = 0 and y=12y = \frac{1}{2}.
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